Hopf-Type Estimates and Formulas For Nonconvex Nonconcave Hamilton--Jacobi break Equations

Simple explicit estimates are presented for the viscosity solution of the Cauchy problem for the Hamilton--Jacobi equation where either the Hamiltonian or the initial data are the sum of a convex and a concave function. The estimates become equalities whenever a "minmax" equals a "maxmin" and thus a representation formula for the solution is obtained, generalizing the classical Hopf formulas as well as some formulas of Kruzkov [Functional Anal. Appl., 2 (1969), pp. 128--136]. ©1998 Society for Industrial and Applied Mathematics